What is the limit as x approaches 0 of (cot(2x))/(csc(x)) ?

The limit as x approaches 0 of (cot(2x))/(csc(x)) is 1/2

limit as x approaches 0 of (cot(2x))/(csc(x))

x \to 0 lim (\frac{cot ( 2 x )}{csc ( x )})

\frac{1}{2}

Decimal

0.5

Solution steps

x \to 0 lim (\frac{cot ( 2 x )}{csc ( x )})

Use the basic trigonometric identity:

\frac{1}{csc ( x )} = sin (x)

= x \to 0 lim (cot (2 x) sin (x))

Rewrite for L'Hopital

= x \to 0 lim (\frac{sin ( x )}{\frac{1}{c o t ( 2 x )}})

Apply L'Hopital's Rule

= x \to 0 lim (\frac{cos ( x )}{sec ^{2} ( 2 x ) \cdot 2})

Plug in the value

x = 0

= \frac{cos ( 0 )}{sec ^{2} ( 2 \cdot 0 ) \cdot 2}

Simplify

\frac{cos ( 0 )}{sec ^{2} ( 2 \cdot 0 ) \cdot 2} : \frac{1}{2}

= \frac{1}{2}

x \to 0 lim (\frac{( 6 ^{x} - 8 ^{x} )}{x})

\int cos (t) + sin (t) d t

n = 1 \sum \infty \frac{1}{ln ( 3 n )}

t an g e n t y = \frac{3}{x}, (4, \frac{3}{4})

d er i v a t i v e y = 96 x^{2} + 72 x + 12

What is the limit as x approaches 0 of (cot(2x))/(csc(x)) ?
The limit as x approaches 0 of (cot(2x))/(csc(x)) is 1/2